September  2020, 13(9): 2403-2424. doi: 10.3934/dcdss.2020120

A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies

1. 

Department of Mathematics, Faculty of Science, Cairo University, Egypt

2. 

Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, 15551, UAE

3. 

Department of Mathematics, Faculty of Education, Sana'a University, Yemen

* Corresponding author: Nasser Sweilam

Received  October 2018 Revised  December 2019 Published  October 2019

In this paper, we present an optimal control problem of fractional-order delay-differential model for cancer treatment based on the synergy between anti-angiogenic and immune cells therapies. The governed model consists of eighteen differential equations. A discrete time-delay is incorporated to represent the time required for the immune system to interact with the cancer cells, and fractional-order derivative is considered to reflect the memory and hereditary properties in the process. Two control variables for immunotherapy and anti-angiogenic therapy are considered to reduce the load of cancer cells. Necessary conditions that guarantee the existence and the uniqueness of the solution for the control problem have been considered. We approximate numerically the solution of the optimal control problem by solving the state system forward and adjoint system backward in time. Some numerical simulations are provided to validate the theoretical results.

Citation: Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120
References:
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O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.  Google Scholar

[2]

J. C. ArcieroT. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discret Contin. Dyn. Syst. Ser. B, 4 (2004), 39-58.  doi: 10.3934/dcdsb.2004.4.39.  Google Scholar

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A. J. ArenasG. González-Parra and B. M. Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional-order, Mathematics and Computers in Simulation, 121 (2016), 48-63.  doi: 10.1016/j.matcom.2015.09.001.  Google Scholar

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D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.  Google Scholar

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A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Applied Mathematical Modeling, 40 (2016), 832-845.  doi: 10.1016/j.apm.2015.06.012.  Google Scholar

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H. M. Byrne, The effect of time-delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.  Google Scholar

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M. A. Cameron and A. L. Davis, A mathematical model of angiogenesis in glioblastoma multiforme, Arizona State University, (2009). Google Scholar

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R. Caponetto, G. Dongola and L. Fortuna, Fractional-Order Systems: Modeling and Control Applications, World Scientific, London, 2010. Google Scholar

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A. R. M. Carvalho and C. M. A. Pinto, Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection, Commun Nonlinear Sci. Numer. Simulat., 61 (2018), 104-126.  doi: 10.1016/j.cnsns.2018.01.012.  Google Scholar

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A. d'OnofrioF. GatiiP. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumor-immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.  Google Scholar

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G. González-ParraA. J. Arena and B. M. Chen-Charpentier, A fractional-order epidemic model for the simulation of outbreaks of influenza A(H1N1), Math. Meth. Appl. Sci., 37 (2014), 2218-2226.  doi: 10.1002/mma.2968.  Google Scholar

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B. JoshiX. Y. WangS. BanerjeeH. Y. TianA. Matzavinos and M. A. J. Chaplain, On immunotherapies and cancer vaccination protocols: A mathematical modeling approach, J. Theor. Biol., 259 (2009), 820-827.  doi: 10.1016/j.jtbi.2009.05.001.  Google Scholar

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B. JoshiX. Y. WangS. BanerjeeH. Y. TianA. Matzavinos and M. A. J. Chaplain, On immunotherapies and cancer vaccination protocols: A mathematical modelling approach, J. Theor. Biol., 259 (2009), 820-827.  doi: 10.1016/j.jtbi.2009.05.001.  Google Scholar

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A. Khan and G. Zaman, Optimal control strategy of SEIR endemic model with continuous age-structure in the exposed and infectious classes, Optim. Control Appl. Meth., 39 (2018), 1716-1727.  doi: 10.1002/oca.2437.  Google Scholar

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Bio., 37 (1998), 235-252.   Google Scholar

[23]

Y. LouzounC. XueG. B. Lesinski and A. Friedman, A mathematical model for pancreatic cancer growth and treatments, J. Theor. Biol., 351 (2014), 74-82.   Google Scholar

[24]

J. A. T. Machado, Fractional-order derivative approximations in discrete-time control systems, Syst. Anal. Model. Simul., 34 (1999), 419-434.   Google Scholar

[25]

C. D. MathersC. Boschi-pinto and A. D. Lopez, Cancer incidence, mortality and survival by site for 14 regions of the world, World Healh Organization, 13 (2001), 1-47.   Google Scholar

[26]

H. Neves and H. Fai Kwok, Recent advances in the field of anti-cancer immunotherapy, BBA Clinical, 3 (2015), 280-288.   Google Scholar

[27]

C. M. A. Pintoa and A. R. M. Carvalho, The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains, Ecological Complexity, 32 (2017), 1-20.   Google Scholar

[28]

M. J. Piotrowska, An immune system-tumour interactions model with discrete time-delay: Model analysis and validation, Commun. Nonlinear Sci. Numer. Simulat., 34 (2016), 185-198.  doi: 10.1016/j.cnsns.2015.10.022.  Google Scholar

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[30]

J. Poleszczuk, P. Hahnfeldt and H. Enderling, Therapeutic implications from sensitivity analysis of tumor angiogenesis models, PLOS One, (2015). Google Scholar

[31]

F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim and M. A. Abdeen, Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control, Comput. Math. Methods Med., (2014), Art. ID 982978, 15 pp. doi: 10.1155/2014/982978.  Google Scholar

[32]

F. A. RihanD. H. AbdelrahmanS. Lakshmanan and A. S. Alkhajeh, A time-delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar

[33]

F. A. Rihan and N. F. Rihan, Dynamics of cancer-immune system with external treatment and optimal control, J. Cancer Sci. Ther., 8 (2016), 257-261.   Google Scholar

[34]

M. Robertson-TessiA. El-Kareh and A. Goriely, A mathematical model of tumor-immune interactions, J. Theor. Biol., 294 (2012), 56-73.  doi: 10.1016/j.jtbi.2011.10.027.  Google Scholar

[35]

P. Bi, S. G. Ruan and X. A. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays, Chaos, 24 (2014), 023101, 16 pp. doi: 10.1063/1.4870363.  Google Scholar

[36]

A. B. SalatiM. Shamsi and D. F. M. Torres, Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 334-350.  doi: 10.1016/j.cnsns.2018.05.011.  Google Scholar

[37]

R. SchererS. L. KallaY. F. Tang and J. F. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput Math Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.  Google Scholar

[38]

L. Soto-Ortiza, A cancer treatment based on synergy between anti-angiogenic and immune cell therapies, Journal of Theoretical Biology, 394 (2016), 197-211.  doi: 10.1016/j.jtbi.2016.01.026.  Google Scholar

[39]

A. Stevens, M. C. Mackey and editors, Mathematical Methods and Models in Biomedicine, Springer, 2013. Google Scholar

[40]

N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optim. Control Appl. Meth., 37 (2016), 1355-1374.  doi: 10.1002/oca.2247.  Google Scholar

[41]

N. H. Sweilam and S. M. AL-Mekhlafi, Comparative study for multi-strain Tuberculosis (TB) model of fractional-order, AMIS, 10 (2016), 1403-1413.   Google Scholar

[42]

N. H.Sweilam and S. M. AL-Mekhlafi, Optimal control for a time delay multi-strain tuberculosis fractional model: A numerical approach, IMA Journal of Mathematical Control and Information, 36 (2019), 317-340.  doi: 10.1093/imamci/dnx046.  Google Scholar

[43]

N. H. Sweilam and S. M. AL-Mekhlafi, Optimal control for a nonlinear mathematical model of tumor under immune suppression: A numerical approach, Optim Control Appl Meth., 39 (2018), 1581-1596.  doi: 10.1002/oca.2427.  Google Scholar

[44]

N. H. Sweilam and S. M. AL-Mekhlafi, Legendre spectral-collocation method for solving fractional optimal control of HIV infection of Cd4+T cells mathematical model, The Journal of Defense Modeling and Simulation, 14 (2017), 273-284.   Google Scholar

[45]

S. Wilson and D. Levy, A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy, Bull Math Biol., 74 (2012), 1485-1500.  doi: 10.1007/s11538-012-9722-4.  Google Scholar

[46]

H. Xu, Analytical approximations for a population growth model with fractional-order, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1978-1983.   Google Scholar

[47]

T. Akman Yildiz, A fractional dynamical model for honeybee colony population, International Journal of Biomathematics, 11 (2018), 1850063, 23 pp. doi: 10.1142/S1793524518500638.  Google Scholar

show all references

References:
[1]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.  Google Scholar

[2]

J. C. ArcieroT. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discret Contin. Dyn. Syst. Ser. B, 4 (2004), 39-58.  doi: 10.3934/dcdsb.2004.4.39.  Google Scholar

[3]

A. J. ArenasG. González-Parra and B. M. Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional-order, Mathematics and Computers in Simulation, 121 (2016), 48-63.  doi: 10.1016/j.matcom.2015.09.001.  Google Scholar

[4]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.  Google Scholar

[5]

A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Applied Mathematical Modeling, 40 (2016), 832-845.  doi: 10.1016/j.apm.2015.06.012.  Google Scholar

[6]

H. M. Byrne, The effect of time-delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.  Google Scholar

[7]

M. A. Cameron and A. L. Davis, A mathematical model of angiogenesis in glioblastoma multiforme, Arizona State University, (2009). Google Scholar

[8]

R. Caponetto, G. Dongola and L. Fortuna, Fractional-Order Systems: Modeling and Control Applications, World Scientific, London, 2010. Google Scholar

[9]

A. R. M. Carvalho and C. M. A. Pinto, Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection, Commun Nonlinear Sci. Numer. Simulat., 61 (2018), 104-126.  doi: 10.1016/j.cnsns.2018.01.012.  Google Scholar

[10]

R. DenysiukC. J. Silva and D. F. M. Torres, Multiobjective optimization to a TB-HIV/AIDS coinfection optimal control problem, Comp. Appl. Math., 37 (2018), 2112-2128.  doi: 10.1007/s40314-017-0438-9.  Google Scholar

[11]

K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn., 71 (2013), 613-619.  doi: 10.1007/s11071-012-0475-2.  Google Scholar

[12]

A. d'OnofrioF. GatiiP. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumor-immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.  Google Scholar

[13]

R. EftimieJ. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull Math Biol., 73 (2011), 2-32.  doi: 10.1007/s11538-010-9526-3.  Google Scholar

[14]

A. M. A. El-SayedA. E. M. El-Mesiry and H. A. A. El-Saka, On the fractional-order logistic equation, Appl. Math. Lett., 20 (2007), 817-823.  doi: 10.1016/j.aml.2006.08.013.  Google Scholar

[15]

H. Enderling and M. A. J. Chaplain, Mathematical modeling of tumor growth and treatment, Curr. Pharm. Des., 20 (2014), 1-7.   Google Scholar

[16]

G. González-ParraA. J. Arena and B. M. Chen-Charpentier, A fractional-order epidemic model for the simulation of outbreaks of influenza A(H1N1), Math. Meth. Appl. Sci., 37 (2014), 2218-2226.  doi: 10.1002/mma.2968.  Google Scholar

[17]

M. J. HayatN. HowladerM. E. Reichman and B. K. Edwards, Cancer statistics, trends, and multiple primary cancer analyses from the Surveillance, Epidemiology, and End Results (SEER) Program, Oncologist, 12 (2007), 20-37.   Google Scholar

[18]

A. JemalM. M. CenterC. Desantis and E. M. Ward, Global patterns of cancer incidence and mortality rates and trends, Cancer Epidemiol biomarkers Prev, 19 (2010), 1893-1907.   Google Scholar

[19]

B. JoshiX. Y. WangS. BanerjeeH. Y. TianA. Matzavinos and M. A. J. Chaplain, On immunotherapies and cancer vaccination protocols: A mathematical modeling approach, J. Theor. Biol., 259 (2009), 820-827.  doi: 10.1016/j.jtbi.2009.05.001.  Google Scholar

[20]

B. JoshiX. Y. WangS. BanerjeeH. Y. TianA. Matzavinos and M. A. J. Chaplain, On immunotherapies and cancer vaccination protocols: A mathematical modelling approach, J. Theor. Biol., 259 (2009), 820-827.  doi: 10.1016/j.jtbi.2009.05.001.  Google Scholar

[21]

A. Khan and G. Zaman, Optimal control strategy of SEIR endemic model with continuous age-structure in the exposed and infectious classes, Optim. Control Appl. Meth., 39 (2018), 1716-1727.  doi: 10.1002/oca.2437.  Google Scholar

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Bio., 37 (1998), 235-252.   Google Scholar

[23]

Y. LouzounC. XueG. B. Lesinski and A. Friedman, A mathematical model for pancreatic cancer growth and treatments, J. Theor. Biol., 351 (2014), 74-82.   Google Scholar

[24]

J. A. T. Machado, Fractional-order derivative approximations in discrete-time control systems, Syst. Anal. Model. Simul., 34 (1999), 419-434.   Google Scholar

[25]

C. D. MathersC. Boschi-pinto and A. D. Lopez, Cancer incidence, mortality and survival by site for 14 regions of the world, World Healh Organization, 13 (2001), 1-47.   Google Scholar

[26]

H. Neves and H. Fai Kwok, Recent advances in the field of anti-cancer immunotherapy, BBA Clinical, 3 (2015), 280-288.   Google Scholar

[27]

C. M. A. Pintoa and A. R. M. Carvalho, The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains, Ecological Complexity, 32 (2017), 1-20.   Google Scholar

[28]

M. J. Piotrowska, An immune system-tumour interactions model with discrete time-delay: Model analysis and validation, Commun. Nonlinear Sci. Numer. Simulat., 34 (2016), 185-198.  doi: 10.1016/j.cnsns.2015.10.022.  Google Scholar

[29] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[30]

J. Poleszczuk, P. Hahnfeldt and H. Enderling, Therapeutic implications from sensitivity analysis of tumor angiogenesis models, PLOS One, (2015). Google Scholar

[31]

F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim and M. A. Abdeen, Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control, Comput. Math. Methods Med., (2014), Art. ID 982978, 15 pp. doi: 10.1155/2014/982978.  Google Scholar

[32]

F. A. RihanD. H. AbdelrahmanS. Lakshmanan and A. S. Alkhajeh, A time-delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606-623.  doi: 10.1016/j.amc.2014.01.111.  Google Scholar

[33]

F. A. Rihan and N. F. Rihan, Dynamics of cancer-immune system with external treatment and optimal control, J. Cancer Sci. Ther., 8 (2016), 257-261.   Google Scholar

[34]

M. Robertson-TessiA. El-Kareh and A. Goriely, A mathematical model of tumor-immune interactions, J. Theor. Biol., 294 (2012), 56-73.  doi: 10.1016/j.jtbi.2011.10.027.  Google Scholar

[35]

P. Bi, S. G. Ruan and X. A. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays, Chaos, 24 (2014), 023101, 16 pp. doi: 10.1063/1.4870363.  Google Scholar

[36]

A. B. SalatiM. Shamsi and D. F. M. Torres, Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 334-350.  doi: 10.1016/j.cnsns.2018.05.011.  Google Scholar

[37]

R. SchererS. L. KallaY. F. Tang and J. F. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput Math Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.  Google Scholar

[38]

L. Soto-Ortiza, A cancer treatment based on synergy between anti-angiogenic and immune cell therapies, Journal of Theoretical Biology, 394 (2016), 197-211.  doi: 10.1016/j.jtbi.2016.01.026.  Google Scholar

[39]

A. Stevens, M. C. Mackey and editors, Mathematical Methods and Models in Biomedicine, Springer, 2013. Google Scholar

[40]

N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optim. Control Appl. Meth., 37 (2016), 1355-1374.  doi: 10.1002/oca.2247.  Google Scholar

[41]

N. H. Sweilam and S. M. AL-Mekhlafi, Comparative study for multi-strain Tuberculosis (TB) model of fractional-order, AMIS, 10 (2016), 1403-1413.   Google Scholar

[42]

N. H.Sweilam and S. M. AL-Mekhlafi, Optimal control for a time delay multi-strain tuberculosis fractional model: A numerical approach, IMA Journal of Mathematical Control and Information, 36 (2019), 317-340.  doi: 10.1093/imamci/dnx046.  Google Scholar

[43]

N. H. Sweilam and S. M. AL-Mekhlafi, Optimal control for a nonlinear mathematical model of tumor under immune suppression: A numerical approach, Optim Control Appl Meth., 39 (2018), 1581-1596.  doi: 10.1002/oca.2427.  Google Scholar

[44]

N. H. Sweilam and S. M. AL-Mekhlafi, Legendre spectral-collocation method for solving fractional optimal control of HIV infection of Cd4+T cells mathematical model, The Journal of Defense Modeling and Simulation, 14 (2017), 273-284.   Google Scholar

[45]

S. Wilson and D. Levy, A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy, Bull Math Biol., 74 (2012), 1485-1500.  doi: 10.1007/s11538-012-9722-4.  Google Scholar

[46]

H. Xu, Analytical approximations for a population growth model with fractional-order, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1978-1983.   Google Scholar

[47]

T. Akman Yildiz, A fractional dynamical model for honeybee colony population, International Journal of Biomathematics, 11 (2018), 1850063, 23 pp. doi: 10.1142/S1793524518500638.  Google Scholar

Figure 1.  Numerical simulations of the state variables with two controls treatment cases and different values of $ d_{\tau} $, $ \alpha = 0.96 $ using IOCM
Figure 2.  Numerical simulations of the control variables and different values of $ d_{\tau} $, $ \alpha = 0.96 $ using IOCM
Figure 3.  Relation between the state variables when $ d_{\tau} = 3 $ and $ \alpha = 0.98 $ using IOCM with two controls treatment cases
Figure 4.  Numerical simulations of the control variables and different values of $ \alpha $ and $ d_{\tau} = 1 $, using IOCM
Figure 5.  The relationship between the variables $ T(t-d_{\tau}), I(t-d_{\tau}), V(t-d_{\tau}) $ and $ R(t-d_{\tau}) $ and $ T(t), I(t), V(t) $ and $ R(t) $ values of $ \alpha = 0.92 $ and $ d_{\tau} = 2 $ using IOCM
Figure 6.  The relationship between the variables T(tdτ), I(tdτ), V (tdτ) and R(tdτ) and T(t), I(t), V (t) and R(t) values of α = 0.92 and dτ = 2 using IOCM
Table 1.  The parameters of system (2)-(19) and their descriptions
Parameters of fraction power Descriptions
$ \gamma^{\alpha} $ The tumor growth rate
$ a^{\alpha} $ The antigenicity for tumor
$ \lambda^{\alpha} $, $ \delta_{u}^{\alpha} $, $ \delta_{D}^{\alpha} $, $ I_{1}^{\alpha} $, $ R_{1}^{\alpha} $ The parameters for dendritic cell expansion
$ \alpha_{A_{1}}^{\alpha} $ Max growth rate for Ang-1
$ \delta_{A_{1}}^{\alpha} $ Degradation rate for Ang-1
$ \alpha_{A_{2}}^{\alpha} $ Max growth rate for Ang-2
$ \delta_{A_{2}}^{\alpha} $ Degradation rate for Ang-2
$ \theta_{A_{2}} $ $ \frac{1}{2} $ max cancer cells needed to trigger $ A_{2} $ production
$ \alpha_{v} $ Rate $ VEGF $ is constantly expressed by glioma cells
$ \alpha_{v_{2}} $ Max growth rate of $ VEGF $ production
$ \delta_{v}^{\alpha} $ Degradation rate of $ VEGF $
$ \alpha_{y}^{\alpha} $ Proliferation rate of endothelial cells
$ \delta_{y}^{\alpha} $ Apototic rate of endothelial cells
$ s^{\alpha} $ Conversion factor from microvessels to $ ECs $
$ \gamma_{B}^{\alpha} $ Max rate microvessels break down to $ ECs $
$ \rho^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to cause regression, growth, etc.
$ \omega^{\alpha} $ Max rate $ ECs $ mature to microvessels
$ \theta_{y}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to keep $ ECs $ alive
$ \theta_{v_{a}}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to induce $ EC $ cell cycle
$ \theta_{EC}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{A2}{A1} $ ratio where $ A2 $ blocks tie-2 receptor from $ A1 $
$ \theta_{B}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{A1}{A2} $ ratio where $ A1 $ matures vessels
$ \tau^{\alpha} $ Binding rate of anti-$ VEGF $ antibody with $ VEGF $
$ \rho_{v_{a}}^{\alpha} $ Degradation rate of anti-$ VEGF $ antibody
$ r_{0}^{\alpha} $, $ k_{2}^{\alpha} $, $ k_{3}^{\alpha} $, $ s_{1}^{\alpha} $ Parameters for tumor progression
$ \alpha_{i}^{\alpha}, $ $ i=1,...,7 $ $ M^{\alpha} $
$ k_{4}^{\alpha}, $ $ C_{1}^{\alpha} $, $ s_{j}^{\alpha} $, $ j=1,2 $, $ \delta_{A}^{\alpha} $, Parameters for T-Cell expansions
$ \delta_{E}^{\alpha} $, $ \delta_{H}^{\alpha} $, $ \delta_{R}^{\alpha} $
$ p_{c}^{\alpha} $, $ p_{1}^{\alpha} $, $ p_{2}^{\alpha} $, $ p_{4}^{\alpha} $, Parameters of IL-2, TGF-$ \beta $ and IL-10 concentrations
$ I_{2}^{\alpha} $, $ s_{4}^{\alpha} $, $ \tau_{c}^{\alpha} $, $ \tau_{s}^{\alpha} $, $ \tau^{\alpha} $
Parameters of fraction power Descriptions
$ \gamma^{\alpha} $ The tumor growth rate
$ a^{\alpha} $ The antigenicity for tumor
$ \lambda^{\alpha} $, $ \delta_{u}^{\alpha} $, $ \delta_{D}^{\alpha} $, $ I_{1}^{\alpha} $, $ R_{1}^{\alpha} $ The parameters for dendritic cell expansion
$ \alpha_{A_{1}}^{\alpha} $ Max growth rate for Ang-1
$ \delta_{A_{1}}^{\alpha} $ Degradation rate for Ang-1
$ \alpha_{A_{2}}^{\alpha} $ Max growth rate for Ang-2
$ \delta_{A_{2}}^{\alpha} $ Degradation rate for Ang-2
$ \theta_{A_{2}} $ $ \frac{1}{2} $ max cancer cells needed to trigger $ A_{2} $ production
$ \alpha_{v} $ Rate $ VEGF $ is constantly expressed by glioma cells
$ \alpha_{v_{2}} $ Max growth rate of $ VEGF $ production
$ \delta_{v}^{\alpha} $ Degradation rate of $ VEGF $
$ \alpha_{y}^{\alpha} $ Proliferation rate of endothelial cells
$ \delta_{y}^{\alpha} $ Apototic rate of endothelial cells
$ s^{\alpha} $ Conversion factor from microvessels to $ ECs $
$ \gamma_{B}^{\alpha} $ Max rate microvessels break down to $ ECs $
$ \rho^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to cause regression, growth, etc.
$ \omega^{\alpha} $ Max rate $ ECs $ mature to microvessels
$ \theta_{y}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to keep $ ECs $ alive
$ \theta_{v_{a}}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to induce $ EC $ cell cycle
$ \theta_{EC}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{A2}{A1} $ ratio where $ A2 $ blocks tie-2 receptor from $ A1 $
$ \theta_{B}^{\alpha} $ $ \frac{1}{2} $ max $ \frac{A1}{A2} $ ratio where $ A1 $ matures vessels
$ \tau^{\alpha} $ Binding rate of anti-$ VEGF $ antibody with $ VEGF $
$ \rho_{v_{a}}^{\alpha} $ Degradation rate of anti-$ VEGF $ antibody
$ r_{0}^{\alpha} $, $ k_{2}^{\alpha} $, $ k_{3}^{\alpha} $, $ s_{1}^{\alpha} $ Parameters for tumor progression
$ \alpha_{i}^{\alpha}, $ $ i=1,...,7 $ $ M^{\alpha} $
$ k_{4}^{\alpha}, $ $ C_{1}^{\alpha} $, $ s_{j}^{\alpha} $, $ j=1,2 $, $ \delta_{A}^{\alpha} $, Parameters for T-Cell expansions
$ \delta_{E}^{\alpha} $, $ \delta_{H}^{\alpha} $, $ \delta_{R}^{\alpha} $
$ p_{c}^{\alpha} $, $ p_{1}^{\alpha} $, $ p_{2}^{\alpha} $, $ p_{4}^{\alpha} $, Parameters of IL-2, TGF-$ \beta $ and IL-10 concentrations
$ I_{2}^{\alpha} $, $ s_{4}^{\alpha} $, $ \tau_{c}^{\alpha} $, $ \tau_{s}^{\alpha} $, $ \tau^{\alpha} $
Table 2.  The Parameter values of system (2)-(19) [38]
$ Parameter $ The value of Parameter
$ a^{\alpha} $ $ (10^{-5} day^{-1})^{\alpha} $
$ C^{\alpha} $ $ (0.3 ng^{-1}mL)^{\alpha} $
$ I_{1}^{\alpha} $ $ (0.4 ng^{-1}mL)^{\alpha} $
$ I_{2}^{\alpha} $ $ (0.75 ng^{-1}mL)^{\alpha} $
$ K_{2}^{\alpha} $ $ (1.2 )^{\alpha} $
$ K_{3}^{\alpha} $ $ (11)^{\alpha} $
$ K_{4}^{\alpha} $ $ (0.33 )^{\alpha} $
$ M^{\alpha} $ $ (10^{7} cell)^{\alpha} $
$ M_{E}^{\alpha} $ $ (3\times10^{6} cell)^{\alpha} $
$ M_{H}^{\alpha} $ $ (6\times10^{6} cell)^{\alpha} $
$ M_{R}^\alpha $ $ (1\times10^{6} cell)^{\alpha} $
$ p_{1}^{\alpha} $ $ (1.8\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{2}^{\alpha} $ $ (1.1\times10^{-7} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{3}^{\alpha} $ $ (1.4\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{4}^{\alpha} $ $ (1.3\times10^{-10} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{c}^{\alpha} $ $ (1.5\times10^{-7} ng^{-1}mLday^{-1} cell^{-1})^{\alpha} $
$ r_{0}^{\alpha} $ $ (0.9 day^{-1})^{\alpha} $
$ R_{1}^{\alpha} $ $ (2\times 10^{7})^{\alpha} $
$ s^{\alpha} $ $ (0.7 \frac{EC}{\mu M})^{\alpha} $
$ S_{1}^{\alpha} $ $ (3.5 ng mL^{-1})^{\alpha} $
$ S_{2}^{\alpha} $ $ (2.9 ng mL^{-1})^{\alpha} $
$ S_{3}^{\alpha} $ $ (1.7 ng mL^{-1})^{\alpha} $
$ S_{4}^{\alpha} $ $ (0.9 ng mL^{-1})^{\alpha} $
$ V_{1}^{\alpha} $ $ (3.5 ng mL^{-1})^{\alpha} $
$ V_{2}^{\alpha} $ $ (2.9 ng mL^{-1})^{\alpha} $
$ V_{3}^{\alpha} $ $ (0.14 ng mL^{-1})^{\alpha} $
$ \alpha_{1}^{\alpha} $ $ (23 day^{-1})^{\alpha} $
$ \alpha_{2}^{\alpha} $ $ (16 day^{-1})^{\alpha} $
$ \alpha_{3}^{\alpha} $ $ (9.9 day^{-1})^{\alpha} $
$ \alpha_{4}^{\alpha} $ $ (1.9 day^{-1})^{\alpha} $
$ \alpha_{5}^{\alpha} $ $ (5.1 day^{-1})^{\alpha} $
$ \alpha_{6}^{\alpha} $ $ (2.1 day^{-1})^{\alpha} $
$ \alpha_{7}^{\alpha} $ $ (0.022 day^{-1})^{\alpha} $
$ \alpha_{A_{1}}^{\alpha} $ $ (0.24 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{A_{2}}^{\alpha} $ $ (1.92 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V}^{\alpha} $ $ (3\times 10^{-6} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V_{2}}^{\alpha} $ $ (3.7\times10^{-2} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{Y}^{\alpha} $ $ (0.198 day^{-1})^{\alpha} $
$ \gamma_{1}^{\alpha} $ $ (2.1 day^{-1})^{\alpha} $
$ \gamma_{B}^{\alpha} $ $ (0.8 day^{-1})^{\alpha} $
$ \theta_{A_{2}}^{\alpha} $ $ (10^{6} cell)^{\alpha} $
$ \theta_{B}^{\alpha} $ $ (1)^{\alpha} $
$ Parameter $ The value of Parameter
$ a^{\alpha} $ $ (10^{-5} day^{-1})^{\alpha} $
$ C^{\alpha} $ $ (0.3 ng^{-1}mL)^{\alpha} $
$ I_{1}^{\alpha} $ $ (0.4 ng^{-1}mL)^{\alpha} $
$ I_{2}^{\alpha} $ $ (0.75 ng^{-1}mL)^{\alpha} $
$ K_{2}^{\alpha} $ $ (1.2 )^{\alpha} $
$ K_{3}^{\alpha} $ $ (11)^{\alpha} $
$ K_{4}^{\alpha} $ $ (0.33 )^{\alpha} $
$ M^{\alpha} $ $ (10^{7} cell)^{\alpha} $
$ M_{E}^{\alpha} $ $ (3\times10^{6} cell)^{\alpha} $
$ M_{H}^{\alpha} $ $ (6\times10^{6} cell)^{\alpha} $
$ M_{R}^\alpha $ $ (1\times10^{6} cell)^{\alpha} $
$ p_{1}^{\alpha} $ $ (1.8\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{2}^{\alpha} $ $ (1.1\times10^{-7} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{3}^{\alpha} $ $ (1.4\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{4}^{\alpha} $ $ (1.3\times10^{-10} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{c}^{\alpha} $ $ (1.5\times10^{-7} ng^{-1}mLday^{-1} cell^{-1})^{\alpha} $
$ r_{0}^{\alpha} $ $ (0.9 day^{-1})^{\alpha} $
$ R_{1}^{\alpha} $ $ (2\times 10^{7})^{\alpha} $
$ s^{\alpha} $ $ (0.7 \frac{EC}{\mu M})^{\alpha} $
$ S_{1}^{\alpha} $ $ (3.5 ng mL^{-1})^{\alpha} $
$ S_{2}^{\alpha} $ $ (2.9 ng mL^{-1})^{\alpha} $
$ S_{3}^{\alpha} $ $ (1.7 ng mL^{-1})^{\alpha} $
$ S_{4}^{\alpha} $ $ (0.9 ng mL^{-1})^{\alpha} $
$ V_{1}^{\alpha} $ $ (3.5 ng mL^{-1})^{\alpha} $
$ V_{2}^{\alpha} $ $ (2.9 ng mL^{-1})^{\alpha} $
$ V_{3}^{\alpha} $ $ (0.14 ng mL^{-1})^{\alpha} $
$ \alpha_{1}^{\alpha} $ $ (23 day^{-1})^{\alpha} $
$ \alpha_{2}^{\alpha} $ $ (16 day^{-1})^{\alpha} $
$ \alpha_{3}^{\alpha} $ $ (9.9 day^{-1})^{\alpha} $
$ \alpha_{4}^{\alpha} $ $ (1.9 day^{-1})^{\alpha} $
$ \alpha_{5}^{\alpha} $ $ (5.1 day^{-1})^{\alpha} $
$ \alpha_{6}^{\alpha} $ $ (2.1 day^{-1})^{\alpha} $
$ \alpha_{7}^{\alpha} $ $ (0.022 day^{-1})^{\alpha} $
$ \alpha_{A_{1}}^{\alpha} $ $ (0.24 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{A_{2}}^{\alpha} $ $ (1.92 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V}^{\alpha} $ $ (3\times 10^{-6} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V_{2}}^{\alpha} $ $ (3.7\times10^{-2} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{Y}^{\alpha} $ $ (0.198 day^{-1})^{\alpha} $
$ \gamma_{1}^{\alpha} $ $ (2.1 day^{-1})^{\alpha} $
$ \gamma_{B}^{\alpha} $ $ (0.8 day^{-1})^{\alpha} $
$ \theta_{A_{2}}^{\alpha} $ $ (10^{6} cell)^{\alpha} $
$ \theta_{B}^{\alpha} $ $ (1)^{\alpha} $
Table 3.  Comparisons between the value of objective functional using IOCM with and without controls cases and $ T_{f} = 100 $, $ d_{\tau} = 2 $
$ \alpha $ $ J(u_A^{\ast}, u_M^{\ast}) $ without control $ J(u_A^{\ast}, u_M^{\ast}) $ with Two controls
1 $ 7.7864 \times 10^{8} $ $ 9.402886\times 10^{4} $
0.90 $ 1.90928 \times 10^{9} $ $ 8.9381\times 10^{4} $
0.80 $ 1.7003\times 10^{9} $ $ 8.1454\times 10^{4} $
0.70 $ 9.0261 \times 10^{8} $ $ 6.7976\times 10^{4} $
0.60 $ 2.1615\times 10^{8} $ $ 4.5080\times 10^{4} $
0.50 $ 1.8270\times 10^{7} $ $ 2.9438\times 10^{4} $
0.40 $ 1.5197\times 10^{6} $ $ 1.9119\times 10^{3} $
0.30 $ 2.5346\times 10^{5} $ $ 855.8715 $
$ \alpha $ $ J(u_A^{\ast}, u_M^{\ast}) $ without control $ J(u_A^{\ast}, u_M^{\ast}) $ with Two controls
1 $ 7.7864 \times 10^{8} $ $ 9.402886\times 10^{4} $
0.90 $ 1.90928 \times 10^{9} $ $ 8.9381\times 10^{4} $
0.80 $ 1.7003\times 10^{9} $ $ 8.1454\times 10^{4} $
0.70 $ 9.0261 \times 10^{8} $ $ 6.7976\times 10^{4} $
0.60 $ 2.1615\times 10^{8} $ $ 4.5080\times 10^{4} $
0.50 $ 1.8270\times 10^{7} $ $ 2.9438\times 10^{4} $
0.40 $ 1.5197\times 10^{6} $ $ 1.9119\times 10^{3} $
0.30 $ 2.5346\times 10^{5} $ $ 855.8715 $
Table 4.  The value of objective functional $ J(u_A^{\ast}, u_M^{\ast}) $, $ T_{f} = 50 $ with different value of $ d_{\tau} $ and control case
$ d_{\tau} $ $ J(u_A^{\ast}, u_M^{\ast}) $
0 $ 4.2401\times 10^4 $
1 $ 4.2419\times 10^4 $
3 $ 4.2447\times 10^4 $
5 $ 4.2467\times 10^4 $
10 $ 4.2504\times 10^4 $
$ d_{\tau} $ $ J(u_A^{\ast}, u_M^{\ast}) $
0 $ 4.2401\times 10^4 $
1 $ 4.2419\times 10^4 $
3 $ 4.2447\times 10^4 $
5 $ 4.2467\times 10^4 $
10 $ 4.2504\times 10^4 $
Table 5.  Comparisons between IOCM, GEM and $ d_{\tau} = 1, $ $ T_{f} = 10. $
$ \alpha $ Methods $ J(u_A^{\ast}, u_M^{\ast}) $
1 IOCM $ 3.2798 \times 10^3 $
GEM $ 8.8514 \times 10^3 $
0.98 IOCM $ 2.6575 \times 10^3 $
GEM $ 1.2839 \times 10^4 $
0.90 IOCM $ 1.1506 \times 10^3 $
GEM $ 7.1531 \times 10^4 $
0.80 IOCM $ 414.5745 $
GEM $ 1.4226\times 10^6 $
$ \alpha $ Methods $ J(u_A^{\ast}, u_M^{\ast}) $
1 IOCM $ 3.2798 \times 10^3 $
GEM $ 8.8514 \times 10^3 $
0.98 IOCM $ 2.6575 \times 10^3 $
GEM $ 1.2839 \times 10^4 $
0.90 IOCM $ 1.1506 \times 10^3 $
GEM $ 7.1531 \times 10^4 $
0.80 IOCM $ 414.5745 $
GEM $ 1.4226\times 10^6 $
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